Question

How do you find the domain and range of `f(x)= sqrt(x^4-16x^2)`?

Answer 1

Domain `{x in RR, x=0, x>=4 or x<=-4}`

Range `{y in RR, y>=0}`

Explanation:

Write `f(x)= sqrt (x^2(x-4)(x+4))`

For f(x) to be real , either `x =0, or (x-4)(x+4) >=0`

This implies that either both x-4 and x+4 should be`>=0 or <=0`
which means either `x>=4 or x<=-4`.

Hence domain would be `{x in RR, x=0, x>=4 or x<=-4}`. In iterval notation it would be `(-oo,-4]U 0 U[4,oo)`

For range it is clear that for x=0, y=0 and for `x>=4 or x<=-4`, y would be positive. Thus range would be `{y in RR, y>=0}`